The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.06 - Nov.-Dec. (2012 vol.9)
pp: 1709-1723
K. Sargsyan , Sandia Nat. Labs., Livermore, CA, USA
C. Safta , Sandia Nat. Labs., Livermore, CA, USA
B. Debusschere , Sandia Nat. Labs., Livermore, CA, USA
H. Najm , Sandia Nat. Labs., Livermore, CA, USA
ABSTRACT
In this work, the problem of representing a stochastic forward model output with respect to a large number of input parameters is considered. The methodology is applied to a stochastic reaction network of competence dynamics in Bacillus subtilis bacterium. In particular, the dependence of the competence state on rate constants of underlying reactions is investigated. We base our methodology on Polynomial Chaos (PC) spectral expansions that allow effective propagation of input parameter uncertainties to outputs of interest. Given a number of forward model training runs at sampled input parameter values, the PC modes are estimated using a Bayesian framework. As an outcome, these PC modes are described with posterior probability distributions. The resulting expansion can be regarded as an uncertain response function and can further be used as a computationally inexpensive surrogate instead of the original reaction model for subsequent analyses such as calibration or optimization studies. Furthermore, the methodology is enhanced with a classification-based mixture PC formulation that overcomes the difficulties associated with representing potentially nonsmooth input-output relationships. Finally, the global sensitivity analysis based on the multiparameter spectral representation of an observable of interest provides biological insight and reveals the most important reactions and their couplings for the competence dynamics.
INDEX TERMS
Response surface methodology, Uncertainty, Stochastic processes, Computational modeling, Computational biology, Bioinformatics,probability and statistics, Approximation, spectral methods
CITATION
K. Sargsyan, C. Safta, B. Debusschere, H. Najm, "Multiparameter Spectral Representation of Noise-Induced Competence in Bacillus Subtilis", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.9, no. 6, pp. 1709-1723, Nov.-Dec. 2012, doi:10.1109/TCBB.2012.107
REFERENCES
[1] N. van Kampen, Stochastic Processes in Physics and Chemistry. Elsevier Science, 1992.
[2] D. Gillespie, Markov Processes: An Introduction for Physical Scientists. Academic Press, 1992.
[3] S. Macnamara, K. Burrage, and R. Sidje, “Multiscale Modeling of Chemical Kinetics via the Master Equation,” Multiscale Modeling and Simulation, vol. 6, no. 4, pp. 1146-1168, 2008.
[4] S. MacNamara, A. Bersani, K. Burrage, and R. Sidje, “Stochastic Chemical Kinetics and the Total Quasi-Steady-State Assumption: Application to the Stochastic Simulation Algorithm and Chemical Master Equation,” J. Chemical Physics, vol. 129, pp. 95-105, 2008.
[5] B. Munsky and M. Khammash, “The Finite State Projection Approach for the Analysis of Stochastic Noise in Gene Networks,” IEEE Trans. Automatic Control, vol. 53, no. 1, pp. 201-214, Jan. 2008.
[6] D. Gillespie, “A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions,” J. Computational Physics, vol. 22, pp. 403-434, 1976.
[7] D. Gillespie, “Exact Stochastic Simulation of Coupled Chemical Reactions,” J. Physical Chemistry, vol. 81, no. 25, pp. 2340-2361, 1977.
[8] R. Gunawan, Y. Cao, L. Petzold, and F.J. Doyle, “Sensitivity Analysis of Discrete Stochastic Systems,” Biophysical J., vol. 88, no. 4, pp. 2530-2540, Apr. 2005.
[9] S. Plyasunov and A. Arkin, “Efficient Stochastic Sensitivity Analysis of Discrete Event Systems,” J. Computational Physics, vol. 221, pp. 724-738, 2007.
[10] M. Komorowski, J. Zurauskiene, and M. Stumpf, “Stochsens - Matlab Package for Sensitivity Analysis of Stochastic Chemical Systems,” Bioinformatics, vol. 28, no. 5, pp. 731-733, 2012.
[11] Y.M. Marzouk, H.N. Najm, and L.A. Rahn, “Stochastic Spectral Methods for Efficient Bayesian Solution of Inverse Probelms,” J. Computational Physics, vol. 224, no. 2, pp. 560-586, 2007.
[12] Y.M. Marzouk and H.N. Najm, “Dimensionality Reduction and Polynomial Chaos Acceleration of Bayesian Inference in Inverse Problems,” J. Computational Physics, vol. 228, no. 6, pp. 1862-1902, 2009.
[13] G.M. Suel, R.P. Kulkarni, J. Dworkin, J. Garcia-Ojalvo, and M.B. Elowitz, “Tunability and Noise Dependence in Differentiation Dynamics,” Science, vol. 315, no. 5819, pp. 1716-1719, 2007.
[14] G. Suel, J. Garcia-Ojalvo, L. Liberman, and M. Elowitz, “An Excitable Gene Regulatory Circuit Induces Transient Cellular Differentiation,” Nature, vol. 440, no. 23, pp. 545-550, 2006.
[15] H. Maamar, A. Raj, and D. Dubnau, “Noise in Gene Expression Determines Cell Fate in Bacillus Subtilis,” Science, vol. 317, no. 5837, pp. 526-529, July 2007.
[16] D. Schultz, E.B. Jacob, J. Onuchic, and P. Wolynes, “Molecular Level Stochastic Model for Competence Cycles in Bacillus Subtilis,” Proc. Nat'l Academy of Sciences USA, vol. 104, no. 45, pp. 17582-17587, 2007.
[17] D. Kim, B. Debusschere, and H. Najm, “Spectral Methods for Parametric Sensitivity in Stochastic Dynamical Systems,” Biophysics J., vol. 92, no. 2, pp. 379-393, Jan. 2007.
[18] D. Sivia, Data Analysis: A Bayesian Tutorial. Oxford Science, 1996.
[19] K. Sanft, S. Wu, M. Roh, J. Fu, R.K. Lim, and L. Petzold, “Stochkit2: Software for Discrete Stochastic Simulation of Biochemical Systems with Events,” Bioinformatics, vol. 27, no. 17, pp. 2457-2458, 2011.
[20] N. Wiener, “The Homogeneous Chaos,” Am. J. Math., vol. 60, pp. 897-936, 1938.
[21] R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach. Springer Verlag, 1991.
[22] M. Rosenblatt, “Remarks on a Multivariate Transformation,” Annals of Math. Statistics, vol. 23, no. 3, pp. 470-472, 1952.
[23] K. Sargsyan, C. Safta, R. Berry, J. Ray, B. Debusschere, and H. Najm, “Efficient Uncertainty Quantification Methodologies for High-Dimensional Climate Land Models,” Sandia Technical Report SAND2011-8757, Nov. 2011.
[24] D.A. Cox, J. Little, and D. O'Shea, Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, 1997.
[25] S.A. Smolyak, “Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions,” Soviet Math. Doklady, vol. 4, pp. 240-243, 1963.
[26] Markov Chain Monte Carlo in Practice, W.R. Gilks, S. Richardson, and D.J. Spiegelhalter, eds., pp. 59-74. Chapman and Hall, 1996.
[27] C. Andrieu, N. de Freitas, A. Doucet, and M.I. Jordan, “An Introduction to MCMC for Machine Learning,” Machine Learning, vol. 50, pp. 5-43, 2003.
[28] O. Le Maître, H. Najm, R. Ghanem, and O. Knio, “Multi-Resolution Analysis of Wiener-Type Uncertainty Propagation Schemes,” J. Computational Physics, vol. 197, pp. 502-531, 2004.
[29] K. Sargsyan, B. Debusschere, H. Najm, and O.L. Maître, “Spectral Representation and Reduced Order Modeling of the Dynamics of Stochastic Reaction Networks via Adaptive Data Partitioning,” SIAM J. Scientific Computing, vol. 31, no. 6, pp. 4395-4421, 2010.
[30] X. Wan and G.E. Karniadakis, “An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations,” J. Computational Physics, vol. 209, pp. 617-642, 2005.
[31] J. Gonzalez, I. Rojas, H. Pomares, J. Artega, and A. Prieto, “A New Clustering Technique for Function Approximation,” IEEE Trans. Neural Networks, vol. 13, no. 1, pp. 132-142, Jan. 2002.
[32] F. Campolongo, A. Saltelli, T. Sørensen, and S. Tarantola, “Hitchhiker's Guide to Sensitivity Analysis,” Sensitivity Analysis, A. Saltelli, K. Chan, and E. Scott, eds., Wiley, 2000.
[33] I.M. Sobol, “Sensitivity Estimates for Nonlinear Mathematical Models,” Math. Modeling and Computational Experiment, vol. 1, pp. 407-414, 1993.
46 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool